Quantum Harmonic Oscillator
Plane Wave
Plane wave in one dimension:
Where: p = momentum, E = Energy, q = position, h = Planks constant, k = spring constant, w = angular frequency and
The expression above represents Eigen functions of operators
Where:
Also:
Where:
Hamiltonian Operator
Multiplying both sides by wave function we get Schrodinger equation:
Since wave function is a variable only we use ordinary differential equations:
Schrodinger Equation in Non-dimensional Form
Consider the variable substitution:
Apply differentiation:
Take the reciprocal of both sides:
Let:
and
Therefore:
We apply this variable substitution to:
Now we substitute those results into Schrodinger equation:
Let:
Therefore:
Then the Schrodinger equation becomes:
Define Schrodinger equation in terms of normalised energy:
Where:
Factorisation of Schrodinger Equation
After a few derivations it can be shown that the factorisation of:
is given by:
We can define this equation in terms of operators b+ and b-:
Where:
Then the Schrodinger equation will equal to:
or
